Obviously, if $f$ is continuous at $x_0$ the statement is right. But do we know if $f$ is continuous at $x_0$, so we can say if the statement is right or wrong?
2026-04-01 07:20:42.1775028042
If $f'$ changes sign on both sides of a specific point $x_0$, then $f$ has an extremum at that point $x_0$.
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It doesn't work in general. For example, $$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 2 - x & \text{if } x \ge 1 \end{cases}.$$ Such a function has a supremum of $2$, but no actual maximum.